Boundary behaviour of harmonic functions in a half-space and brownian motion

Burkholder, D. L.; Gundy, Richard F.

Annales de l'Institut Fourier, Tome 23 (1973), p. 195-212 / Harvested from Numdam

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Abstract

Soit $u (x, y)$ une fonction harmonique dans le demi-espace $R_{+}^{n + 1}$ , $n \geq 2$ . Nous montrons que $u (x, y)$ peut avoir une limite fine en presque chaque point du cube unité dans $R^{n} = \partial R_{+}^{n + 1}$ sans avoir pourtant de limite non tangentielle en aucun point du cube. La méthode est probabiliste et utilise l’équivalence entre limites conditionnelles du mouvement brownien et limites fines à la frontière.

Dans $R_{+}^{2}$ , il est connu que l’on peut caractériser les classes de Hardy $H^{p}$ , $0 < p < \infty$ , par l’intégrabilité de la fonction maximale du mouvement brownien. Nous montrons que ce résultat est aussi valable dans $R_{+}^{n + 1}$ , pour $n \geq 2$ .

Let $u$ be harmonic in the half-space $R_{+}^{n + 1}$ , $n \geq 2$ . We show that $u$ can have a fine limit at almost every point of the unit cubs in $R^{n} = \partial R_{+}^{n + 1}$ but fail to have a nontangential limit at any point of the cube. The method is probabilistic and utilizes the equivalence between conditional Brownian motion limits and fine limits at the boundary.

In $R_{+}^{2}$ it is known that the Hardy classes $H^{p}$ , $0 < p < \infty$ , may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms of the integrability of a Brownian motion maximal function. This result is shown to hold in $R_{+}^{n + 1}$ , for $n \geq 2$ .

Publié le : 1973-01-01

MR 51 #1943 | Zbl 0253.31010 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.487

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     author = {Burkholder, D. L. and Gundy, Richard F.},
     title = {Boundary behaviour of harmonic functions in a half-space and brownian motion},
     journal = {Annales de l'Institut Fourier},
     volume = {23},
     year = {1973},
     pages = {195-212},
     doi = {10.5802/aif.487},
     mrnumber = {51 \#1943},
     zbl = {0253.31010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1973__23_4_195_0}
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Burkholder, D. L.; Gundy, Richard F. Boundary behaviour of harmonic functions in a half-space and brownian motion. Annales de l'Institut Fourier, Tome 23 (1973) pp. 195-212. doi : 10.5802/aif.487. http://gdmltest.u-ga.fr/item/AIF_1973__23_4_195_0/

Bibliographie

[1] J. M. Brelot and L. Doob, Limites angulaires et limites fines, Ann. Inst. Fourier (Grenoble), 13, (1963), 395-415. | Numdam | MR 33 #4299 | Zbl 0132.33902

[2] D.L. Burkholder and R.F. Gundy, Distribution function inequalities for the area integral, Studia Math., 44, (1972), 527-544. | MR 49 #5309 | Zbl 0219.31009

[3] D.L. Burkholder, R.F. Gundy and M. L. Silverstein, A maximal function characterization of the class Hp, Trans. Amer. Math. Soc., 157 (1971), 137-153. | MR 43 #527 | Zbl 0223.30048

[4] A. P. Calderón, On the behaviour of harmonic functions at the boundary, Trans. Amer. Math. Soc., 68, (1950), 47-54. | MR 11,357e | Zbl 0035.18901

[5] A. P. Calderón, On a theorem of Marcinkiewicz and Zygmund, Trans. Amer. Math. Soc., 68, (1950), 55-61. | MR 11,357f | Zbl 0035.18903

[6] L. Carleson, On the existence of boundary values for harmonic functions in several variables, Arkiv för Mathematik, 4, (1961), 393-399. | MR 28 #2232 | Zbl 0107.08402

[7] C. Constantinescu and A. Cornea, Über das Verhalten der analytischen Abildungen Riemannscher Flachen auf dem idealen Rand von Martin, Nagoya Math. J., 17, (1960), 1-87. | MR 23 #A1025 | Zbl 0104.29901

[8] J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France, 85, (1957), 431-458. | Numdam | MR 22 #844 | Zbl 0097.34004

[9] J. L. Doob, Boundary limit theorems for a half-space, J. Math. Pures Appl., (9) 37, (1958), 385-392. | MR 22 #845 | Zbl 0097.34101

[10] C. Fefferman and E. M. Stein, Hp-spaces in several variables, Acta Math., 129, (1972), 137-193. | MR 56 #6263 | Zbl 0257.46078

[11] G. H. Hardy and J. E. Littlewood, Some properties of conjugate functions, J. fur Mat., 167, (1931), 405-423. | JFM 58.0333.03 | Zbl 0003.20203

[12] J. Lelong, Review 4471, Math. Reviews, 40, (1970), 824-825.

[13] J. Lelong, Étude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup., 66, (1949), 125-159. | Numdam | Zbl 0033.37301

[14] J. Marcinkiewicz and A. Zygmund, A theorem of Lusin, Duke Math. J., 4, (1938), 473-485. | JFM 64.0268.01 | Zbl 0019.42001

[15] H. P. Mckean Jr., Stochastic integrals, Academic Press, New York, 1969. | Zbl 0191.46603

[16] L. Naïm, Sur le rôle de la frontière de R. S. Martin dans la Théorie du potential, Ann. Inst. Fourier (Grenoble), 7, (1957), 183-285. | Numdam | MR 20 #6608 | Zbl 0086.30603

[17] I. I. Privalov, Integral Cauchy, Saratov, 1919.

[18] D. Spencer, A function-theoretic identity, Amer. J. Math., 65, (1943), 147-160. | MR 4,137f | Zbl 0060.20603

[19] E. M. Stein, On the theory of harmonic functions of several variables II. Behaviour near the boundary, Acta Math., 106, (1961), 137-174. | MR 30 #3234 | Zbl 0111.08001

[20] E. M. Stein and G. Weiss, On the theory of harmonic functions of several variables I. The theory of Hp-spaces, Acta Math., 103, (1960), 25-62. | MR 22 #12315 | Zbl 0097.28501

[21] J. L. Walsh, The approximation of harmonic functions by harmonic polynomials and harmonic rational functions, Bull. Amer. Math. Soc., 35, (1929), 499-544. | JFM 55.0889.05